Salesforce: Process Builder, Flow and Workflow running in System Mode

System mode: process running is NOT DEPEND on the current user permissions to the object access, field level security, sharing rule. In which the level permissions of the current user are ignored.

User mode: process running is DEPEND on the current user permission to the object access, field level security, sharing rule. In which the level permissions of the current user are enforced.

Automation in Salesforce will run in System mode, in the declarative mode this is include: Workflow, Process Builder, Process Builder + Flow. The same applicable for apex trigger as well.

While Flow will run in User mode, this is make sense as Flow is not count as automation, Flow have user interface that need user to fill something or click button.

Thanks to Mayank Srivastava for sharing matrix below this in Success Community.

The matrix is tested as of Summer '16 release, so this may change in the future. 

When Flow hit error if user do not have access to the field, an error email will be sent to the Flow creator:
Subject: Error Occurred During Flow "Flow_Name": INSERT --- INSERT FAILED --- ERRORS : (INVALID_FIELD_FOR_INSERT_UPD...

Body: An error occurred at element Element_Name (FlowRecordCreate).
INSERT --- INSERT FAILED --- ERRORS : (INVALID_FIELD_FOR_INSERT_UPDATE) Unable to create/update fields: Field_Name__c. Please check the security settings of this field and verify that it is read/write for your profile or permission set., 

Reference:

• Process Builder Security Context
• Getting Started with Process Builder